We define an effective divisor on $\bR^N$ to be a function with finite support $\mu:\bR^N\to\bZ_{\geq 0}$. Its mass, denoted by $\bm(\mu)$, is the nonnegative integer

$$\bm(\mu)=\sum_{\bp\in\bR^N} \mu(\bp). $$

We denote by $\Div_+(\bR^N)$ the set of effective divisors. Note that $\Div_+(\bR^N)$ has a natural structure of Abelian semigroup.

For any $\bp\in\bR^N$ we denote by $\delta_\bp$ the Dirac divisor of mass $1$ and supported at $\bp$. The Dirac divisors generate the semigroup $\Div_+(\bR^N)$. We have a natural topology on $\Div_+(\bR^N)$ where $\mu_n\to \mu$ if and only if

Step 3. (Replacement) We will show that for any distinct points $\bq_1,\bq_2$ and any positive integers $m_1,m_2$ we can find $(m_1+m_2)$ equidistant points $\bp_1,\dotsc,\bp_{m_1+m_2}$ on the line determined by $\bq_1$ and $\bq_2$ such that

This is elementary. Without restricting the generality we can assume that $\bq_1$ and $\bq_2$ lie on an axis (or geodesic) $\bR$ of $\bR^N$, $\bq_0=0$ and $\bq_2=q>0$. Clearly we can find real numbers $x_0, r$, $r>0$, such that
$$
\frac{1}{m_1}\sum_{j=1}^{m_1}(x_0+j r)=0,\;\;\frac{1}{m_2}\sum_{j=m_1+1}^{m_1+m_2}(x_0+jr)=q. $$

If $\bc_0(\mu_1)=\bc_0(\mu_2)$ the divisors $\eC_0(\mu_1)$ $\eC_0(\mu_2)$ are supported at the same point and we are done. Suppose that $\bq_1=\bc_0(\mu_1)\neq\bc_0(\mu_2)=\bq_2$. By Step 3, we can find equidistant points $\bp_1,\dotsc,\bp_{m_1+m_2}$ such that

Remark. The above proof does not really use the linear structure. If we uses only the fact that any two points in $\bR^N$ determine a unique geodesic. The Normalization condition can be replaced by the equivalent one